Stochastic iterative learning control of continuous linear system

CRONE control of continuous linear time periodic systems: Application to a testing bench

This paper develops a hybrid iterative learning control (ILC) algorithm which uses both tracking and contour errors as correction signals during the learning process. The tracking, contour and combined ILCs are first introduced and analyzed. A hybrid ILC is then proposed to predict contour errors based on the identified models and then used to determine whether the tracking or contour errors should be utilized for the correction signals. The criteria for selecting different signals are based on the predicted root mean square (RMS) value of the contour errors for non‐uniform rational B‐spline (NURBS) curves. Simulations performed on a butterfly trajectory represented by NURBS curves show that the hybrid ILC adopts tracking errors as correction signals for the first five iterations, even if the contour error RMS values are used as the objective function. As the iterations evolve, the hybrid ILC switches from using tracking errors to using contour errors as correction signals. Geometric interpretations are given to illustrate the switching behavior. It is shown that the hybrid ILC can significantly reduce contour errors as compared to the process without learning. Finally, validation experiments using the butterfly curve show that the hybrid ILC outperforms the tracking, contour and combined ILCs algorithms.Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Model Reduction via Generalized Frequency Interval Cross Gramian

This paper is concerned with the variable initial states problem in iterative learning control (ILC) for linear continuous systems. Firstly, the properties of the trajectory of 2-D continuous-discrete Roesser model are analyzed by using Lyapunov's method. Then, for any variable initial states which absolutely converge to the desired initial state, some sufficient conditions in the form of linear matrix inequalities (LMI) are given to ensure the convergence of the PD-type ILC rules. It implies that the ILC rules can be used to achieve the perfect tacking for variable initial states, even if the system dynamic is unknown. Finally, two numerical examples are given to illustrate the perfect tracking performance with exponentially convergent initial states.

ABSTRACTIn this article, to tackle with the iteration-varying trail lengths and random initial state shifts, an average operator-based PD-type iterative learning control (ILC) law is firstly presented for linear discrete-time multiple-input multiple-output (MIMO) systems with vector relative degree. The proposed PD-type ILC law includes an initial rectifying action against initial state shifts, and pursues the reference trajectory tracking beyond the initial time points. As special cases of the PD-type ILC law, P-type and D-type ILC laws are then introduced. It is proved that for linear discrete-time MIMO systems with vector relative degree, the three proposed ILC laws can drive the varying trail lengths-based ILC tracking errors to zero in mathematical expectation beyond the initial time points. A numerical example is used to illustrate the effectiveness of the proposed ILC laws.

O044 Changes in upper limb isometric strength and error tracking following training using functional electrical stimulation

The epsilon-entropy was adopted as a measure of information in the analysis of linear Gaussian continuous time control systems under the consideration that the `virtual reproductions' of system input and output are subjected to a common high enough precision requirement. The function of system variety which describes the variation of time average information in system was defined as the difference between epsilon-entropy rates of system input and output. It formulates how does the information or uncertainty of input transmit through the dynamic system and impact output. Relations within variety, Bode sensitivity integral and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sub> entropy were then derived.

A linear continuous time-varying system and a sample-data system with continuous time-varying linear part are considered. For both systems a two-dimensional control is formed which provides an invariant stabilization of a closed loop system. I. INTRODUCTION The invariant stabilization problem deals with such stabi- lization which provides an independence of output from outer action and boundedness of other variables. As demonstrated in (1-3) this problem remains one of the most actual in the theory of control systems. The main conclusion of these papers is: beginning from linear stationary systems the scalar control can provide either stability or invariance. A statement in (4) says that for a linear stationary system it is possible to combine the requirements of invariance and stability if the outer action is considered as one of the regulator's input. In the present report a linear continuous time-varying system and a sample-data system with continuous time- varying linear part are considered. For both systems a two- dimensional control is formed which provides an invariant stabilization of a closed loop system. If the vector of ob- servation is orthogonal to the vector of outer action the measurement of the action is not required. II. LINEAR NONSTATIONARY SYSTEM The following system

Sain and Massey [1] have obtained necessary and sufficient conditions for the invertibility of continuous and discrete time linear systems and have also found bounds for what they termed the "inherent integration" (continuous time) or the "inherent delay" (discrete time) of an invertible linear system. In this note, we tighten these bounds by modifying an argument used to prove the Sain-Massey result.

This paper presents a novel model reference adaptive iterative learning control (ILC) for unknown continuous-time linear time-varying systems. The unknown time-varying parameters of the system are neither required to vary slowly nor to have known bounds. The system is not required to be minimum-phase, stable, controllable or observable. The input of the system is determined by a differentiator-free control law. The used reference model is time-invariant and first order and thus choosing its parameters is easily possible, even though, the system under control is high order and time variant. Almost all of the components of the system initial condition can be iteration variant. By introducing a novel kind of Lyapunov function the convergence of the proposed adaptive ILC (AILC) and achieving asymptotic tracking are proved. Also, by rigorous mathematical analysis and with the help of some mathematical key techniques such as Bellman-Gronwall lemma, it is shown that all signals and quantities in the closed-loop system are bounded in the sense of at least one norm. Finally, the effectiveness of the proposed method is verified by two simulation examples.

The tracking performance of systems that perform repetitive tasks can be significantly improved using iterative learning control (ILC). During successive iterations, ILC learns a high performance feedforward signal from the measured tracking error. In practical applications, the tracking errors of successive experiments contain a repetitive part and a non‐repetitive part. ILC only compensates for the repetitive part, while the non‐repetitive part also enters the learning scheme and deteriorates the performance of ILC. In this paper, analysis of the tracking error of ILC shows the influence of non‐repetitive disturbances. The disturbances of the last two iterations appear to have the largest influence on the tracking error. In order to remove the non‐repetitive disturbances from the tracking error, a wavelet filtering method is proposed, which identifies and removes the non‐repetitive disturbances by a comparison of the time–frequency content of two error realizations for each iteration of ILC. The wavelet filtered error signal contains only the repetitive disturbances and is used as input for ILC. Both simulations and experiments show that with wavelet filtering, a better tracking performance is obtained together with a feedforward signal that contains significantly less disturbances. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Traditional and typical iterative learning control algorithm shows that the convergence rate of error is very low for a class of regular linear systems. A fast iterative learning control algorithm is designed to deal with this problem in this paper. The algorithm is based on the traditional P-type iterative learning control law, which increases the composition of adjacent two overlapping quantities, the tracking error of previous cycle difference signals, and the current error difference. Using convolution to promote Young inequalities proved strictly that, in terms of Lebesgue-p norm, when the number of iterations tends to infinity, the tracking error converges to zero in the system and presents the convergence condition of the algorithm. Compared with the traditional P-type iterative learning control algorithm, the proposed algorithm improves convergence speed and evades the defect using the norm metric’s tracking error. Finally, the validation of the effectiveness of the proposed algorithm is further proved by simulation results.

Iterative learning control (ILC) is applicable to systems that are required to repeatedly track a desired trajectory of finite duration. Norm-optimal ILC can be characterised as a combined feedforward and feedback learning approach, where the tracking error from the previous trial and the tracking error of the current trial are employed to reduce the tracking error from trial to trial. In this paper, a high speed rack feeder typically used in automated warehouses is considered, which represents a flexible beam structure with a vertically moving mass. Due to kinematic constraints such as a maximum velocity and a maximum acceleration, standard ILC is not applicable if the desired trajectory violates these constraints. One possible solution would be an offline trajectory planning subject to the given kinematic constraints. This paper, however, addresses modifications of the ILC algorithm itself to cope with infeasible trajectories. Two alternative algorithms are given for this purpose and compared with each other in experiments on a test rig that replicates the dynamics of a high speed rack feeder.

This note considers the problem of iterative learning control (ILC) for a class of linear continuous-time multiple-input multiple-output (MIMO) systems with vector relative degree, where the trail length of control input is different from that of system state and output for a specific iteration. An iteration-average operator is included in the proposed ILC law to address the varying trail lengths. The proposed ILC algorithm allows that the control input length is iteration-varying during the ILC process, and can achieve desired trajectory tracking at control time interval. A numerical example is carried out to illustrate the effectiveness of the proposed ILC scheme.

In this paper, iterative learning control (ILC) is considered to solve the tracking problem of time-varying linear stochastic systems with randomly varying trial lengths. Using the two-dimensional Kalman filtering technique, the authors can establish a recursive framework for designing the learning gain matrix along both time and iteration axes by optimizing the trace of input error covariance matrix. It is strictly proved that the input error converges to zero asymptotically in mean square sense and thus the tracking error covariance converges. The extensions to that prior distribution of nonuniform trial lengths is unknown are also investigated with an asymptotical estimation method. Numerical simulations are provided to verify the effectiveness of the proposed framework.